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| | {{unsolved|mathematics| Does there exist any number that can be expressed as a sum of 2 positive ''5''th powers in at least 2 different ways, i.e., ''a''<sup>5</sup> + ''b''<sup>5</sup> <nowiki>=</nowiki> ''c''<sup>5</sup> + ''d''<sup>5</sup>?}} |
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| | In [[mathematics]], the '''generalized taxicab number''' ''Taxicab''(''k'', ''j'', ''n'') is the smallest number which can be expressed as the sum of ''j'' ''k''th positive powers in ''n'' different ways. For ''k'' = 3 and ''j'' = 2, they coincide with [[Taxicab number]]s. |
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| | It has been shown by [[Euler]] that |
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| | :<math>\mathrm{Taxicab}(4, 2, 2) = 635318657 = 59^4 + 158^4 = 133^4 + 134^4.</math> |
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| | However, ''Taxicab''(5, 2, ''n'') is not known for any ''n'' ≥ 2; no positive [[integer]] is known which can be written as the sum of two fifth powers in more than one way.<ref>{{cite book |
| | | last = Guy |
| | | first = Richard K. |
| | | authorlink = Richard K. Guy |
| | | title = Unsolved problems in number theory (third edition) |
| | | publisher = Springer-Science+Business Media, Inc. |
| | | date = 2004 |
| | | location = New York, New York, USA |
| | | pages = 437 |
| | | url = http://books.google.com/books?id=1AP2CEGxTkgC&printsec=frontcover#v=onepage&q=&f=false |
| | | doi = |
| | | id = |
| | | isbn = 0-387-20860-7 }} |
| | </ref> |
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| | It can be easily verified on a home computer using a simple brute force search that the ''Taxicab''(5, 2, ''2'') problem has no solutions with { ''a, b, c, d'' } all less than 1,000. A more in-depth search shows the same is true for all combinations up to 4,000. A lower bound on the solution is |
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| | :<math>\mathrm{Taxicab}(5, 2, 2) > 1,024,000,000,000,000,000 = 1.024 * 10^{18}.</math> |
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| | ==See also== |
| | *[[Cabtaxi number]] |
| | |
| | ==References== |
| | <references /> |
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| | ==External links== |
| | *[[Walter Schneider (mathematician)|Walter Schneider]]:[http://web.archive.org/web/20050425023736/http://www.wschnei.de/number-theory/taxicab-numbers.html Taxicab numbers] |
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| | [[Category:Number theory]] |
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| | [[de:Taxicab-Zahl#Verallgemeinerte Taxicab-Zahl]] |
Endocrinologist Grippo from Capreol, likes to spend time backgammon, property developers in singapore and creating a house. Has lately finished a trip to China Danxia.
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In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number which can be expressed as the sum of j kth positive powers in n different ways. For k = 3 and j = 2, they coincide with Taxicab numbers.
It has been shown by Euler that
However, Taxicab(5, 2, n) is not known for any n ≥ 2; no positive integer is known which can be written as the sum of two fifth powers in more than one way.[1]
It can be easily verified on a home computer using a simple brute force search that the Taxicab(5, 2, 2) problem has no solutions with { a, b, c, d } all less than 1,000. A more in-depth search shows the same is true for all combinations up to 4,000. A lower bound on the solution is
See also
References
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External links
de:Taxicab-Zahl#Verallgemeinerte Taxicab-Zahl